Group #19157482 — subgroups

Subgroups — C₂₃×#9856 #19157482

Order 5888.

530 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1492	C₂	1	4	yes	maximal, complemented
4	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#135	—	2	2	no	—
6	C₂₃×#532	C₂²	1	2	yes	—
7	C₂₃×#495	—	2	2	no	—
8	C₂₃×#494	C₄	1	1	yes	characteristic
9	C₂₃×#493	C₄	1	1	yes	characteristic
10	C₂₃×#492	C₄	1	2	yes	—
11	C₂×C₂₃×#108	—	2	2	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×D₄⋊C₈	—	2	8	no	—
16	C₈²×C₂₃	—	2	2	no	—
17	C₂×#41×C₂₃	—	2	2	no	—
18	C₂₃×#135	—	4	4	no	—
19	C₂₃×#135	—	4	4	no	—
20	C₂×Q₁₆×C₂₃	—	2	2	no	—
21	C₂×SD₁₆×C₂₃	—	2	4	no	—
22	C₂×D₈×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	D₄	1	2	yes	—
26	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	C₂₃×#127	C₂×C₄	1	2	yes	—
28	C₂₃×#124	—	2	2	no	—
29	C₂₃×#121	—	2	4	no	—
30	C₄×D₄×C₂₃	—	2	4	no	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×#108	—	4	4	no	—
33	C₂₃×C₄⋊C₈	—	2	4	no	—
34	C₄×C₈×C₂₃	—	2	2	no	—
35	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₂₃	D₄	1	4	yes	—
38	C₄×C₈×C₂₃	D₄	1	2	yes	—
39	#41×C₂₃	—	4	4	no	—
40	#41×C₂₃	—	4	4	no	—
41	C₂×Q₈×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	—	2	2	no	—
43	C₂×D₄×C₂₃	—	2	4	no	—
44	C₂×D₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	#41	1	2	yes	—
48	Q₁₆×C₂₃	—	4	4	no	—
49	SD₁₆×C₂₃	—	4	8	no	—
50	D₈×C₂₃	—	4	4	no	—
51	M₁₆×C₂₃	—	4	4	no	—
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₂₃	D₈	1	4	yes	—
58	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
59	C₄⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	4	4	no	—
61	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₂×D₄	1	2	yes	—
64	#9856	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	—	4	2	no	—
66	C₂³×C₂₃	#124	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	4	no	—
68	D₄×C₂₃	—	4	4	no	—
69	D₄×C₂₃	—	4	8	no	—
70	D₄×C₂₃	—	8	4	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	4	2	no	—
73	C₂×C₄×C₂₃	—	8	2	no	—
74	C₂×C₄×C₂₃	—	8	2	no	—
75	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₂₃	#121	1	2	yes	—
78	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
81	C₈×C₂₃	—	2	8	no	—
82	C₈×C₂₃	—	8	2	no	—
83	C₈×C₂₃	#102	1	4	yes	—
84	C₈×C₂₃	#102	1	4	yes	—
85	#3147	C₂×C₂₃	1	1	yes	characteristic
86	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
87	#1492	C₂×C₂₃	1	4	yes	complemented
88	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	8	2	no	—
91	C₂²×C₂₃	—	8	4	no	—
92	C₂²×C₂₃	#390	1	1	yes	characteristic
93	C₂²×C₂₃	#486	1	2	yes	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	8	2	no	—
96	C₄×C₂₃	C₂×#102	1	2	yes	—
97	C₄×C₂₃	#443	1	4	yes	—
98	C₄×C₂₃	#492	1	1	yes	characteristic
99	C₄×C₂₃	#493	1	1	yes	characteristic
100	C₂×#135	—	2	2	no	—
101	#532	C₂²×C₂₃	1	2	yes	—
102	#495	—	2	2	no	—
103	#494	C₄×C₂₃	1	1	yes	characteristic
104	#493	C₄×C₂₃	1	1	yes	characteristic
105	#492	C₄×C₂₃	1	2	yes	—
106	C₂×#108	—	2	2	no	—
107	#423	C₂²×C₂₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
110	D₄⋊C₈	—	2	8	no	—
111	C₈²	—	2	2	no	—
112	C₂×C₂₃	—	2	2	no	—
113	C₂×C₂₃	—	16	2	no	—
114	C₂×C₂₃	#1957	1	1	yes	characteristic
115	C₂×C₂₃	#1974	1	2	yes	—
116	C₂×#41	—	2	2	no	—
117	#135	—	4	4	no	—
118	#135	—	4	4	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×SD₁₆	—	2	4	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	D₄×C₂₃	1	2	yes	—
125	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
126	#127	C₂×C₄×C₂₃	1	2	yes	—
127	#124	—	2	2	no	—
128	#121	—	2	4	no	—
129	C₄×D₄	—	2	4	no	—
130	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
131	#108	—	4	4	no	—
132	C₄⋊C₈	—	2	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₄×C₈	D₄×C₂₃	1	4	yes	—
137	C₄×C₈	D₄×C₂₃	1	2	yes	—
138	C₂₃	#9856	1	1	yes	characteristic, direct product factor, Sylow
139	#41	—	4	4	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	#41×C₂₃	1	2	yes	—
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
156	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
157	C₂×C₈	D₈×C₂₃	1	4	yes	—
158	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
162	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
163	C₄²	C₂×D₄×C₂₃	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂₃×#124	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	8	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
176	C₂×C₄	C₂₃×#121	1	2	yes	—
177	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
179	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	8	2	no	—
182	C₈	C₂₃×#102	1	4	yes	—
183	C₈	C₂₃×#102	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	4	no	—
187	C₂²	C₂₃×#390	1	1	yes	characteristic
188	C₂²	C₂₃×#486	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×C₂₃×#102	1	2	yes	—
192	C₄	C₂₃×#443	1	4	yes	—
193	C₄	C₂₃×#492	1	1	yes	characteristic
194	C₄	C₂₃×#493	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	C₂₃×#1957	1	1	yes	characteristic
198	C₂	C₂₃×#1974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#3147

C₂

yes

maximal, characteristic

C₂₃×#1848

C₂

yes

maximal, characteristic, complemented

C₂₃×#1492

C₂

yes

maximal, complemented

C₂×C₈²×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#135

—

C₂₃×#532

C₂²

yes

—

C₂₃×#495

—

C₂₃×#494

C₄

yes

characteristic

C₂₃×#493

C₄

yes

characteristic

C₂₃×#492

C₄

yes

—

C₂×C₂₃×#108

—

C₂₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×D₄⋊C₈

—

C₈²×C₂₃

—

C₂×#41×C₂₃

—

C₂₃×#135

—

C₂₃×#135

—

C₂×Q₁₆×C₂₃

—

C₂×SD₁₆×C₂₃

—

C₂×D₈×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂₃×#127

C₂×C₄

yes

—

C₂₃×#124

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#108

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

Q₁₆×C₂₃

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

D₈

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

—

#9856

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

#124

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄×D₄

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

C₂×D₈

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#102

yes

—

C₈×C₂₃

#102

yes

—

#3147

C₂×C₂₃

yes

characteristic

#1848

C₂×C₂₃

yes

characteristic, complemented

#1492

C₂×C₂₃

yes

complemented

C₂×C₈²

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#390

yes

characteristic

C₂²×C₂₃

#486

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×#102

yes

—

C₄×C₂₃

#443

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

100

—

101

#532

C₂²×C₂₃

yes

—

102

#495

—

103

#494

C₄×C₂₃

yes

characteristic

104

#493

C₄×C₂₃

yes

characteristic

105

#492

C₄×C₂₃

yes

—

106

C₂×#108

—

107

#423

C₂²×C₂₃

yes

—

108

C₂×C₄×C₈

C₂²×C₂₃

yes

—

109

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

110

D₄⋊C₈

—

111

C₈²

—

112

C₂×C₂₃

—

113

C₂×C₂₃

—

114

C₂×C₂₃

#1957

yes

characteristic

115

C₂×C₂₃

#1974

yes

—

116

C₂×#41

—

117

#135

—

118

#135

—

119

C₂×Q₁₆

—

120

C₂×SD₁₆

—

121

C₂×D₈

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

D₄×C₂₃

yes

—

125

C₄⋊Q₈

C₂×C₄×C₂₃

yes

—

126

#127

C₂×C₄×C₂₃

yes

—

127

#124

—

128

#121

—

129

C₄×D₄

—

130

C₂×C₄²

C₂³×C₂₃

yes

characteristic

131

#108

—

132

C₄⋊C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₂×C₄×C₂₃

yes

characteristic

135

C₄×C₈

C₂×C₄×C₂₃

yes

characteristic

136

C₄×C₈

D₄×C₂₃

yes

—

137

C₄×C₈

D₄×C₂₃

yes

—

138

C₂₃

#9856

yes

characteristic, direct product factor, Sylow

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

147

C₂²×C₄

#41×C₂₃

yes

—

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

156

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

157

C₂×C₈

D₈×C₂₃

yes

—

158

C₂×C₈

SD₁₆×C₂₃

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²×C₄×C₂₃

yes

characteristic

162

C₄²

C₂×D₄×C₂₃

yes

characteristic

163

C₄²

C₂×D₄×C₂₃

yes

—

164

C₂³

—

165

C₂³

C₂₃×#124

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

175

C₂×C₄

C₄×D₄×C₂₃

yes

—

176

C₂×C₄

C₂₃×#121

yes

—

177

C₂×C₄

C₂₃×#127

yes

characteristic

178

C₂×C₄

C₂×D₈×C₂₃

yes

—

179

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

180

C₈

—

181

C₈

—

182

C₈

C₂₃×#102

yes

—

183

C₈

C₂₃×#102

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₂₃×#390

yes

characteristic

188

C₂²

C₂₃×#486

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×C₂₃×#102

yes

—

192

C₄

C₂₃×#443

yes

—

193

C₄

C₂₃×#492

yes

characteristic

194

C₄

C₂₃×#493

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₂₃×#1957

yes

characteristic

198

C₂

C₂₃×#1974

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

#127

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

socle

C₂²×C₂₃

#390

prime radical

C₂₃×#3147

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#1848	C₂	semidirect
3. C₂₃×#1492	C₂	semidirect
4. C₂×C₈²×C₂₃	C₂	semidirect
64. #9856	C₂₃	direct (N × H)
86. #1848	C₂×C₂₃	semidirect
87. #1492	C₂×C₂₃	semidirect
88. C₂×C₈²	C₂×C₂₃	semidirect
138. C₂₃	#9856	direct (N × H)

H ≅ G/N

type

2. C₂₃×#1848

C₂

semidirect

3. C₂₃×#1492

C₂

semidirect

4. C₂×C₈²×C₂₃

C₂

semidirect

64. #9856

C₂₃

direct (N × H)

86. #1848

C₂×C₂₃

semidirect

87. #1492

C₂×C₂₃

semidirect

88. C₂×C₈²

C₂×C₂₃

semidirect

138. C₂₃

#9856

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3147	2
C₂₃×#1848	2
C₂₃×#1492	2
C₂×C₈²×C₂₃	2
#9856	23

subgroup

index

C₂₃×#3147

C₂₃×#1848

C₂₃×#1492

C₂×C₈²×C₂₃

#9856

Subgroups — C₂₃×#9856 #19157482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#9856 #19157482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9856 #19157482